First order of business, check connections (sorry if offensive, but weirder things have happened; I had a cable which went bad that mimicked problems with an amp).
Second, get a replacement tube. I'm not an expert on tube construction, but it is possible in my thinking that an element of the tube expands upon heating up to a certain point and the contact is lost. Swap the two tubes and see if the problem moves to the other channel. If so, it's a tube issue. Be cautious and purposeful moving the tubes; get the pins lined up and push down carefully so as not to bend or break them. You can do it, just take your time.
Swapping the tubes should not hurt the unit; operate it in low volume as you are testing and have a finger on the mute button in case. These are extreme precautions which are likely not necessary, but I always take extra precautions when diagnosing and changing things with amps. If you have not yet exchanged tubes I would guess it might be the tube gone kaput.
Have you conducted tests to make sure it's not the source?
If none of this resolves the issue it's possible there is a solder joint which is opening up when the unit heats up. A tech should be able to find it and fix it fairly easily.
If still a problem contact Pathos to find a tech nearby. You an email Gianni Borinato at Pathos - he's the designer of the MkIII. If anyone on earth has an idea of it, it's him. Send the email to pathos' headqarters with his name and he'll get it.